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Difficulty: Very hardPolygon Angles and Properties

A designer is creating a custom tiled floor using irregular convex pentagonal tiles. In each pentagon, the measures of three of the interior angles are in the ratio 2:3:42:3:4. The other two interior angles are congruent to each other, and each is 15∘15^\circ less than the sum of the two smallest angles in the ratio. What is the measure, in degrees, of the largest interior angle of one of these pentagonal tiles?

  1. A
    120∘120^\circ
  2. 135∘135^\circAnswer
  3. C
    150∘150^\circ
  4. D
    165∘165^\circ
  5. E
    105∘105^\circ

Answer

The correct answer is 135 degrees. The largest interior angle of the pentagon is one of the two congruent angles.
The correct answer is 135 degrees. The sum of the interior angles of a pentagon is 540 degrees. Representing the three angles in the ratio as 2x, 3x, and 4x gives a sum of 9x. The remaining two angles are each equal to the sum of the two smallest ratio terms minus 15, which is 5x - 15. The sum of all five angles is 19x - 30 = 540, which yields x = 30. Evaluating the angles gives 60, 90, 120, 135, and 135 degrees. The largest of these is 135 degrees.

Step-by-Step Solution

1
Determine the sum of the interior angles of a pentagon.
The sum is 540∘540^\circ.
The formula for the sum of the interior angles of an nn-sided polygon is (n−2)×180∘(n-2) \times 180^\circ. For a pentagon (n=5n = 5), the sum is (5−2)×180∘=3×180∘=540∘(5-2) \times 180^\circ = 3 \times 180^\circ = 540^\circ.
2
Set up an algebraic equation representing the sum of all five interior angles.
The equation is 2x+3x+4x+2(5x−15∘)=540∘2x + 3x + 4x + 2(5x - 15^\circ) = 540^\circ.
Let the three angles in the ratio 2:3:42:3:4 be represented as 2x2x, 3x3x, and 4x4x. The sum of the two smallest is 2x+3x=5x2x + 3x = 5x. Each of the other two congruent angles is 15∘15^\circ less than this sum, which is 5x−15∘5x - 15^\circ.
3
Solve the algebraic equation for xx.
x=30∘x = 30^\circ.
Simplify the equation: 9x+10x−30∘=540∘  ⟹  19x=570∘  ⟹  x=30∘9x + 10x - 30^\circ = 540^\circ \implies 19x = 570^\circ \implies x = 30^\circ.
4
Calculate the measures of all five interior angles and identify the largest.
The angles are 60∘60^\circ, 90∘90^\circ, 120∘120^\circ, 135∘135^\circ, and 135∘135^\circ. The largest angle is 135∘135^\circ.
Substitute x=30∘x = 30^\circ into each expression: 2(30)=60∘2(30) = 60^\circ, 3(30)=90∘3(30) = 90^\circ, 4(30)=120∘4(30) = 120^\circ, and 5(30)−15=135∘5(30) - 15 = 135^\circ for the other two. The largest value among these is 135∘135^\circ.

Key Concept

Polygon interior angle sum and algebraic representation of ratios
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Topics

Polygon Angles and Properties

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